# Fibonacci proof by induction

## Homework Statement

$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

## The Attempt at a Solution

P(k+1):$$F_{2k-1}+F_{2k+1}$$=$$F_{2k+2}$$

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What happened to F_1 and F_3?

What happened to F_1 and F_3?
well, it's F_1+F_3+....+F_2n-1

Then P(k + 1) needs to be changed accordingly.

What have you tried so far?

I've tried plugging in numbers.

And how did that work out?

The definition of Fibonacci numbers will be helpful in the induction proof.

If I plug in 1, I just get F_1, so 1=1
If I plug in 2, I get F_3, so 1+2=F_4, 3=3

well, it's F_1+F_3+....+F_2n-1
F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...

HallsofIvy
Homework Helper

## Homework Statement

$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

## The Attempt at a Solution

P(k+1):$$F_{2k-1}+F_{2k+1}$$=$$F_{2k+2}$$
well, it's F_1+F_3+....+F_2n-1
What was "F_1+ F_3+ ...+ F_2n-1"?

In your first post you said the problem was to prove that
$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

Are you saying now it is actually to prove that
$$F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}$$=$$F_{2n}$$?

F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...
Very close. What happens when 2 consecutive Fibonacci numbers are added?

Very close. What happens when 2 consecutive Fibonacci numbers are added?
It equals the 3rd Fibonnacci number.
F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1

And that's exactly what you wanted to show.

Ok thanks!

epenguin
Homework Helper
Gold Member
Sorry but maybe the problem is not properly stated? Are the F defined to be Fibonacci numbers?

Then F2n = F2n-1 + F2n-2

So if you are then asking also that

F2n = F2n-1 + F1 + F3

then

F2n-2 = F1 + F3

which is not making much sense.